Wavelet transform modulus: phase retrieval and scattering

Abstract

We discuss the problem that consists in reconstructing a function from the modulus of its wavelet transform. In the case where the wavelets are Cauchy wavelets, all analytic functions are uniquely determined by this modulus. Additionally, although it is not uniformly continuous, the reconstruction operator enjoys a form of local stability. We describe these two results, and give an idea of the proof of the first one. To conclude, we present a related result on a more sophisticated operator, based on the wavelet transform modulus: the scattering transform.